Kubo Formulae for Second-Order Hydrodynamic Coefficients
Guy D. Moore and Kiyoumars A. Sohrabi
Department of Physics, McGill University, 3600 rue University, Montréal QC H3A 2T8, Canada
(Dated: August 18, 2010)
At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity η
and on five additional “second-order” hydrodynamical coefficients τΠ , κ, λ1 , λ2 , and λ3 . We derive
Kubo relations for these coefficients, relating them to equilibrium, fully retarded 3-point correlation
functions of the stress tensor. We show that the coefficient λ3 can be evaluated directly by Euclidean
means and does not in general vanish.
bative behavior of the coefficient λ3 (which is not zero)
and to say something about its physical interpretation.
INTRODUCTION
Results from the RHIC experiments, particularly the
measurement of a large transverse flow [? ], appear to
show that the Quark-Gluon plasma can be well described
by hydrodynamics with a surprisingly small viscosity [?
]. A major future goal for heavy ion experiment and theory is to quantify how small the viscosity of the plasma
is. This requires the numerical treatment of relativistic
viscous hydrodynamics [? ]. It has long been known [?
? ? ] that the relativistic Navier-Stokes equations are
acausal and unstable. But the Navier-Stokes equations
are just the result of a first-order (Chapman-Enskog [?
]) expansion in gradients. Extending the expansion to
second order yields numerically stable equations [? ].
The drawback is that it adds unknown coefficients. In
the conformal case (which we will consider for simplicity), besides the equation of state P (ǫ) at zero order and
the shear viscosity η at first order, there are five new
transport coefficients: τΠ , κ, λ1 , λ2 , λ3 in the notation of
[? ]. These have been evaluated in strongly coupled N =4
SYM theory with infinite colors [? ? ] and at leading
order in weakly coupled QCD [? ]. Both calculations
found λ3 = 0.
Baier et al have also presented Kubo formulae for
two of these coefficients, τΠ and κ, which relate them
to well defined, equilibrium correlation functions of the
stress tensor. Presumably, the remaining three coefficients λ1,2,3 can also be expressed in terms of stress tensor
correlation functions. Doing so would put the definition
of these coefficients on a solid footing and might aid in
their physical interpretation and their theoretical calculation. In the remainder of this paper we will derive such
Kubo relations for the three remaining second-order coefficients. We do this first by showing how the first and
second order hydrodynamic coefficients can be related to
the stress tensor in a background spacetime with perturbatively small geometrical curvature. Then we expand in
the metric as an external background field a la Kubo [?
] and derive a relation between λ1,2,3 and certain fully
retarded 3-point stress-tensor correlation functions. This
allows us to determine the previously unknown pertur-
CONSTITUTIVE RELATIONS FOR SECOND
ORDER COEFFICIENTS
We begin by defining the second order coefficients.
The stress-energy tensor of a fluid can be decomposed
in terms of an equilibrium and an extra piece,[? ]
µν
µν
Tnoneq
= Teq
(uµ , ǫ) + Πµν ,
µν
Teq
≡ (ǫ + P )uµ uν + P g µν .
(1)
The decomposition is unique if we choose the LandauLifshitz convention uµ Πµν = 0 (and u2 = −1). The
key idea of hydrodynamics is that, for a system which
varies slowly in space and time, Πµν should arise only due
to the nonuniformity of the system and should therefore
be expressible in terms of a gradient expansion in that
nonuniformity. To simplify the discussion, we further
specialize to fluids in conformal theories, in which case
Πµµ = 0. In this case, the most general form compatible
with conformal symmetry is [? ]
∇ · u µν
σ
Πµν = −ησ µν + ητΠ h u · ∇σ µνi +
3
hµνi
αhµνiβ
+κ R
− 2uα uβ R
(2)
+λ1 σ hµ λ σ νiλ + λ2 σ hµ λ Ωνiλ + λ3 Ωhµ λ Ωνiλ ,
where we have followed the notation of [? ] by defining
the transverse projection operator ∆µν ≡ g µν + uµ uν
and the space-projection and traceless symmetrization
of indices,
Ahµνi ≡
1 µα νβ
1
∆ ∆ (Aαβ + Aβα ) − ∆µν ∆αβ Aαβ , (3)
2
3
and introduced the shear tensor σ µν and vorticity tensor
Ωµν , defined as
σ µν ≡ 2∇hµ uνi ,
Ω
µν
≡
1 µα νβ
∆ (∇α uβ
2∆
(4)
− ∇β uα ) .
(5)
2
EXPANSION IN BACKGROUND GEOMETRY
We derive Kubo relations for λ1 etc. by considering
a system where some nonuniformity, either in the initial
conditions or in the spacetime geometry, forces σ µν etc.
to be nonzero. It is particularly convenient to consider
an initially uniform, equilibrium system but to introduce
perturbatively weak and slowly varying spacetime deformations which then force the fluid to experience shear
and vorticity. One then expands perturbatively in the
metric deviation hµν . Since a metric distortion hµν couples to the stress tensor T µν , this generates an expansion
in correlation functions of multiple stress tensors, whose
coefficients are the response of the stress tensor to fluid
nonuniformities.
Consider in general the expectation value hT µν (0)i
for a system initially in equilibrium at temperature T ,
subject to a spacetime dependent metric perturbation
hαβ (x), starting at time t0 < 0. The stress tensor is
determined by
Z 0
hT µν (0)i = Tr e−βH T̄exp
dt′ iH[h(t′ )] T µν
Z
1
d4 xd4 yGµν,αβ,γδ
(0, x, y)hαβ (x)hγδ (y)
raa
8
+O(h3 ) ,
(9)
+
where Gµν,...,αβ
(0, . . . , x) is the correlation function of
r...a
one Tr and 0 or more Ta ’s,
(−i)n−1 (−2i)n ∂ n W µν,...,αβ
Gr...a
(0, . . . , x) ≡
(10)
∂ga,µν (0) . . . ∂gr,αβ (x) gµν =ηµν
= (−i)n−1 Trµν (0) . . . Taαβ (x) + c.t.
These functions are fully retarded; that is, they are timeordered, nested commutators, eg when t1 < t2 < . . . < 0
the correlator is h[T (t1 ), [T (t2 ), [. . . T (0)]]]i (the r index
is always at the latest time and innermost in the commutators). Here (c.t.) refers to the contact terms which are
built into our definition of the n-point stress tensor correlation functions. This is discussed in [? ]; the contact
terms turn out not to be important for evaluating η but
they will contribute to the evaluation of the second-order
nonlinear transport coefficients.
t0
×Texp
Z
0
t0
dt (−i)H[h(t )] .
′′
′′
KUBO FORMULAE
(6)
This is best handled with using the Schwinger-Keldysh
(closed time path) formalism (see [? ] and [? ], whose
conventions we will follow). We introduce independent
metric perturbations for the time-ordered and anti-timeordered evolution operators in the above expression and
define the generating functional
Z ∞
dt′ H[h2 (t′ )]
W [h1 , h2 ] ≡ ln Tr e−βH T̄exp i
t
Z ∞0
dt′ H[h1 (t′ )]
×Texp −i
t0
Z
R√
−g1 d4 xL[Φ1 (x),h1 ]
i
= ln D[Φ1 , Φ2 ]e
R√
−g2 d4 yL[Φ2 (y),h2 ]
.
(7)
×e−i
One then defines the average metric perturbation hr ≡
h1 +h2
2
and stress tensor Tr ≡ T1 +T
, and the difference
2
2
variables ha ≡ h1 − h2 , Ta ≡ T1 − T2 . Variation with
respect to ha gives Tr , explicitly
∂W
−2i
√
= Trµν (x) .
−g ∂haµν (x)
(8)
We use such a variation to pull down the T µν (0) factor we
want to evaluate. Now hrµν is the background geometry;
varying order by order in hrµν gives a series expansion of
Trµν in powers of h. Explicitly, we find
Z
1
µν
µν
d4 xGµν,αβ
(0, x)hαβ (x)
hT ih = Gr (0) −
ra
2
First we review the derivation of Kubo formulae for
the “linear” transport coefficients η, τΠ , κ [? ]. Consider
hT xy i in the presence of hxy (z, t). According to Eq. (??),
at first order
Z
xy
hT ih = − d4 x hxy (x)Gxy,xy
(0, x) + O(h2 ) . (11)
ra
Using Eq. (??) and ∇µ T µν = 0 we find ui = 0 at O(h),
but σ xy and some other terms in Eq. (??) are nonzero at
first order in h. Explicitly evaluating σ xy and the other
allowed terms in Eq. (??) in terms of hxy and inserting
into Eq. (??), we find
hT xy ih = −P hxy − η∂t hxy + ητΠ ∂t2 hxy
κ
− ∂z2 hxy + ∂t2 hxy + O(∂ 3 , h2 ) . (12)
2
R
defining Gxy,xy
(ω, k) = d4 xei(ωt−kz) Gxy,xy
(0, x) and
ra
ra
equating Eqs. (??,??) order by order in derivatives, we
find
η = −i∂ω Gxy,xy
(ω, k)|ω=0=k ,
ra
(13)
(ω, k)|ω=0=k ,
(14)
κ = −∂k2z Gxy,xy
ra
1 2 xy,xy
(ω, k) ω=0=k .(15)
∂ G
(ω, k) − ∂k2z Gxy,xy
ητπ =
ra
2 ω ra
These reproduce the Kubo relations obtained by [? ] except that our ω is their −ω, since we Fourier transformed
Gra (0, x) rather than Gra (x, 0).
To obtain higher order Kubo formulae for the nonlinear
coefficients, we continue this procedure to the next order
3
in h, for a background choice which allows nonzero shear
flow and vorticity. For this purpose it is sufficient to consider a background with nonzero hxy (t, z) and nonzero
hx0 (y). In this case we find ui = 0 at the O(h) level, but
σ xy = ∂t hxy and Ωxy = −∂y hx0 /2 at O(h). Explicitly
evaluating Eq. (??) in this background to second order,
we find
4
4
Πxx = ηhxy ∂t hxy + ητΠ − hxy ∂t 2 hxy + ∂t hxy ∂y h0x
3
3
1
κ
2
− (∂t hxy ) +
hxy ∂z 2 hxy + 2hxy ∂t 2 hxy
3
3
1
2
−∂y h0x ∂t hxy − h0x ∂y 2 h0x + λ1 (∂t hxy )
3
1
1
− λ2 ∂t hxy ∂y h0x + λ3 (∂y h0x )2 .
(16)
2
12
Equating with the second-order part of Eq. (??), and
defining
Z
Gµν,αβ,σλ
(p,
q)
≡
d4 xd4 ye−i(p·x+q·y) Gµν,αβ,σλ
(0, x, y)
raa
raa
The gauge change which eliminates hµν is xµ → xµ + ξ µ
with ξµ,ν + ξν,µ = hµν . Applying the gauge change to
the lefthand side of Eq. (??), we re-express it in terms of
hµν and the flat-space correlation functions;
η µγ Gδν,αβ
(p) + (µ ↔ ν)
ra
+ η αγ Gµν,δβ
(p)
+
(α
↔
β)
+
(γ
↔
δ)
ra
hγδ
= 2hγδ Gµν,αβ,γδ
(p, 0) .
raa
For our case µν = xx, αβ = xy = γδ so
2Gxy,xy
(p) + Gxx,yy
(p) + Gxx,xx
(p) = 2Gxx,xy,xy
(p, 0) .
ra
ra
ra
raa
(26)
η
and
Now ∂ω Gxy,xy = iη (Eq. (??)), ∂ω Gxx,xx = 4i
3
η
(which
are
derived
by
the
same
proce∂ω Gxx,yy = −2i
3
dure as Eq. (??)), so Eq. (??) reproduces Eq. (??). The
same procedure applies for the other linear coefficients.
(17)
DISCUSSION
we find the following Kubo relations:
∂2
3
Gxx,xy,xy (p, q)
(18)
lim
2 p,q→0 ∂p0 ∂q 0 raa
2κ
∂2
xx,xy,0x
= 2ητΠ −
Graa
(p, q) (19)
+ 2 lim
p,q→0 ∂p0 ∂q 2
3
∂2
xx,0x,0x
Graa
(p, q) .
(20)
= −6 lim
p,q→0 ∂p2 ∂q 2
λ1 = ητΠ −
λ2
λ3
These Kubo relations are our main result.
We also find the following extra Kubo relations for the
coefficients determined at first order in h:
∂ xx,xy,xy
4iη
Graa
(p, q) ,
(21)
= lim
p,q→0 ∂p0
3
−κ
∂2
Gxx,xy,xy
(p, q) , (22)
= lim
raa
p,q→0 (∂p3 )2
3
4ητΠ − 2κ
∂2
Gxx,xy,xy
(p, q) . (23)
= lim
raa
p,q→0 (∂p0 )2
3
At first sight this is surprising; either the coefficients
are overdetermined, or there are inter-relations between
stress tensor two point functions and three point functions. But each extra Kubo formula involves one stress
tensor at zero external 4-momentum, arising from an undifferentiated hµν in Eq. (??). But we can always force
hµν = 0 at x = 0 where T xx is evaluated by a coordinate “gauge” choice. The invariance of the theory to
such gauge choice enforces (Ward) relations between two
point functions and three point functions with a Taµν at
zero 4-momentum. Consider a stress tensor two-point
function in a spacetime independent background hµν ,
hTrµν (0)Taαβ (x)ih = iGµν,αβ
(0, x)
(24)
ra
Z
i
d4 y hγδ (y)Gµν,αβ,γδ
(0, x, y) .
−
raa
2
(25)
The goal of our derivation was twofold. First, we
wanted relations, shown in Eqs. (??,??,??), for the
second-order nonlinear transport coefficients in terms
of equilibrium energy-momentum tensors. Second, we
hoped that these relations would shed some light on
the nature or properties of these transport coefficients.
The most mysterious of these transport coefficients is λ3 ,
which is found to vanish identically in N =4 SYM theory in the limit of many colors and large coupling [? ]
and which also vanishes at leading perturbative order in
weak coupling [? ]. Is it identically zero? Romatschke
studied this problem (among others) using a generalized
entropy current and showed that its value is related to a
certain modification of the entropy density in the presence of vorticity [? ]. Our Kubo relation allows for a
direct evaluation of λ3 in weakly coupled field theory.
Two coefficients – κ and λ3 – have expressions in
terms of space, but not time, derivatives of stress tensor correlation functions. That means that we may immediately set ω = 0 in Eq. (??) and p0 , q 0 = 0 in
Eq. (??). The frequency-domain, fully retarded n-point
correlation function is the analytic continuation of the
frequency-domain, Euclidean-time ordered[? ] correlation function. In particular, Gµν,αβ,στ
(−iω1 , −iω2 ) =
raa
(ω
,
ω
)
for
Matsubara
frequencies
ω1,2 =
in0 Gµν,αβ,στ
1
2
E
2πT n1,2 . Here n0 is the number of indices µ, ν, α, β, σ, τ
which are 0, since there is a factor of i arising from the
Euclidean continuation of a 0 index. This relation shows
that the zero-frequency raa and Euclidean correlation
functions are equal up to factors of i. Hence
∂2
xx,0x,0x
GE
(p, q) .
p
~,~
q→0 ∂py ∂qy
λ3 = 6 lim
(27)
4
One usually considers such Euclidean correlation functions to carry only thermodynamical information; λ3
should not be thought of as a dynamical coefficient but
as a thermodynamic response to vorticity.[? ]
At weak coupling we can directly evaluate Eq. (??) diagrammatically in the Matsubara formalism. This contrasts with the case of η, τΠ , λ1 , and λ2 , where time
derivatives mean that Graa must be evaluated at small
nonzero frequency where the continuation cannot be so
simply applied. This is why, at weak coupling, the other
coefficients can involve inverse powers of the coupling [?
? ], but κ [? ] and λ3 do not.
We have evaluated the correlation function in Eq. (??)
for a one-component scalar field theory at leading order in weak coupling. Two diagrams contribute; a trian2
gle diagram, ∂py ∂qy hT xx (−p − q)T x0 (p)T x0 (q)i = −T
48
and a contact term involving X x0x0 ≡ 2∂T x0 /∂gx0,
2
∂py ∂qy hT xx (−p − q)X x0x0 (p + q)i = T36 . Other contact
terms involving ∂T xx/∂gx0 , ∂ 2 T xx /(∂gx0 )2 are either p
or q independent and vanish when we apply ∂py ∂qy . Combining, we find
λ3 =
T2
,
24
1 real scalar field
(28)
(at weak coupling). The important observations are that
λ3 can be quite easily evaluated at weak coupling via
Euclidean techniques, and the result is not in general
zero.
Acknowledgements
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
We would like to thank Alessandro Cerioni, Paul Romatschke, Omid Saremi, and Dam Son for useful conversations. This work was supported in part by the Natural
Sciences and Engineering Research Council of Canada.
[14]
[15]
[16]
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Our metric convention is ηµν = Diag[ − + + + ].
Euclidean time ordering is the operator ordering which
naturally arises in the conventional Euclidean path integral.
For instance, there are curved but time-independent geometries where κ and λ3 contribute to T µν but the fluid
is in equilibrium and no entropy production is occurring.